Preparations

Load the necessary libraries

library(car)       #for regression diagnostics
library(broom)     #for tidy output
library(ggfortify) #for model diagnostics
library(DHARMa)    #for residual diagnostics
library(performance) #for residuals diagnostics
library(see)         #for plotting residuals
library(sjPlot)    #for outputs
library(knitr)     #for kable
library(effects)   #for partial effects plots
library(ggeffects) #for partial effects plots
library(emmeans)   #for estimating marginal means
library(modelr)    #for auxillary modelling functions
library(tidyverse) #for data wrangling

Scenario

Here is an example from Fowler, Cohen, and Jarvis (1998). An agriculturalist was interested in the effects of fertilizer load on the yield of grass. Grass seed was sown uniformly over an area and different quantities of commercial fertilizer were applied to each of ten 1 m2 randomly located plots. Two months later the grass from each plot was harvested, dried and weighed. The data are in the file fertilizer.csv in the data folder.

FERTILIZER YIELD
25 84
50 80
75 90
100 154
125 148
... ...
FERTILIZER: Mass of fertilizer (g.m-2) - Predictor variable
YIELD: Yield of grass (g.m-2) - Response variable

The aim of the analysis is to investigate the relationship between fertilizer concentration and grass yield.

Read in the data

fert = read_csv('../data/fertilizer.csv', trim_ws=TRUE)
## Parsed with column specification:
## cols(
##   FERTILIZER = col_double(),
##   YIELD = col_double()
## )
glimpse(fert)
## Rows: 10
## Columns: 2
## $ FERTILIZER <dbl> 25, 50, 75, 100, 125, 150, 175, 200, 225, 250
## $ YIELD      <dbl> 84, 80, 90, 154, 148, 169, 206, 244, 212, 248
## Explore the first 6 rows of the data
head(fert)
str(fert)
## tibble [10 × 2] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
##  $ FERTILIZER: num [1:10] 25 50 75 100 125 150 175 200 225 250
##  $ YIELD     : num [1:10] 84 80 90 154 148 169 206 244 212 248
##  - attr(*, "spec")=
##   .. cols(
##   ..   FERTILIZER = col_double(),
##   ..   YIELD = col_double()
##   .. )

Exploratory data analysis

Model formula: \[ y_i \sim{} \mathcal{N}(\mu_i, \sigma^2)\\ \mu_i = \beta_0 + \beta_1 x_i \]

where \(y_i\) represents the \(i\) observed values, \(\beta_0\) and \(\beta_1\) represent the intercept and slope respectively, and \(\sigma^2\) represents the estimated variance.

ggplot(fert, aes(y=YIELD, x=FERTILIZER)) +
  geom_point() +
  geom_smooth()

ggplot(fert, aes(y=YIELD, x=FERTILIZER)) +
  geom_point() +
  geom_smooth(method='lm')

ggplot(fert, aes(y=YIELD)) +
  geom_boxplot(aes(x=1))

Fit the model

fert.lm<-lm(YIELD~1+FERTILIZER, data=fert)
fert.lm<-lm(YIELD~FERTILIZER, data=fert)

Model validation

Model outputs

Model investigation / hypothesis testing

Predictions

Additional analyses

Summary figures

References

Fowler, J., L. Cohen, and P. Jarvis. 1998. Practical Statistics for Field Biology. England: John Wiley & Sons.